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8/3/2019 Multiuser Detection for OCDMA Systems
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1801IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. 21314, FEBRUARYMARCWAPRIL 1994
Multiuser Detection for Optical Code DivisionMultiple Access SystemsMajitC Brandt-Pearce, Member; IEEE, and Behnaam Aazhang, Senio r Member; IEEE
Abstract -This paper considersa multiuser detection scheme foroptical direct sequence code division multiple access (OCDMA)systems, referred to as the multistage detector. Previous worksin this area have proposed two detectors: the correlation detectorthat is simple, but has poor performancefor large number of users,and the optimal (minimum probability of error) detector that hasexponential complexity in the number of users. Efficient multiuserdetection algorithms consider the interfering user codes at the ex-pense of electronic speed processing, unlike the optical processingachievable with the correlation detector. The multiple access sys-tem using the proposed multistage detector is shown to be of highperformance and low complexity, compared to the conventionalcorrelation detector and the optimal detector. The model studiedincludes multiple access interference as well as the Poisson charac-teristics of the optical direct detection process. An approximationto the probability of error is derived for the multistage detector,and it is compared to the actual error probability of the correlationdetector, to a lower bound to the error probability of the optimaldetector, and to simulation results for the multistage detector. Theprobability of error is calculated using a characteristic functionmethod. Results are presented for a random code case that show asignificant mprovement n the performanceof the OCDMA systemusing this detector over the correlation detector.
I. INTRODUCTIONOptical code division multiple access (OCDMA) communi-cations provides a means to combine the vast bandwidth avail-able in direct detection optical systems with the advantages ofspread spectrum multiplexing. CDMA is a bandwidth utiliza-tion scheme in which many users access a common channelsimultaneously and asynchronously through the use of encod-ing. Each user employs a unique code to differentiate hat userssignal from that of the other users. The encoding process spreadsthe user signal in the frequency domain so that each user occu-pies the entire bandwidth available at all times. This resultsin a total bandwidth utilization and a graceful degradation asthe number of users increases, as the presence of other users
is detrimental only while those users access the channel. Theoptical medium 1s superbly suited to spread spectrum multipleaccess communications due to its extremely large bandwidth.Paper approvedby CostasN. Georghiades, he Editor for Synchronization ndOptical Detection of the JEEE Communications Society. Manuscript receivedJanuary 9, 1992: revised August 4,1992. This work is supported in part byNASNJohnson Space Center under grant NGT-50447 and by the AdvancedTechnologyProgramof the Texas Higher Education Coordinating Board underGrant 003604-018. This paper was presented in part at the Conference onInformation Sciences and Systems, Baltimore, MD, March 1991.M. Brandt-Pearce is with the Department of Electrical Engineering, Univer-
sitv of Vireinia. Charlottesville. VA.B. Aazhang is with the Department of Electrical and Computer Engineering,Rice University, Houston, TX.IEEELog Number 9401582. 0090-6778/94$04.000 1994IEEE
As a multiple user optical communication scheme, OCDMAis an alternative to wavelength division multiple access(WDMA), which requires each user to use a different wave-length laser to transmit. When the number of users is large orthe channel access is bursty, OCDMA offers an advantage overWDMA, due to the OCDMA characteristics of graceful degra-dation as the number of users increase and constant utilizationof the entire bandwidth. Additional advantages of CDMA areflexibility to user allocation, security against unauthorizedusers,and anti-jamming capabilities. As the communication channelis not subdivided into time or frequency slots, CDMA allowsrandom access to an indefinite number of users. As additionalusers subscribe o the system, they can be given unique codes andthen access the channel without the need to synchronize withany other user. Unauthorized users attempting to eavesdropwill find the signal unintelligible unless the codes are known.CDMA systems are also resistant against jamming as a highpower jamming signal would have to cover a significantportionof the CDMA bandwidth to create any noticeable degradation.Due to these advantages, CDMA has been the focus of muchresearch in the last fifteen years, primarily in the radio frequencydomain, but also in the optical domain. Current efforts in the areaof direct detection optical CDMA have focused on performanceevaluations of systems employing a conventional correlationdetector (matched filter) or an optimal detector. Hui proposedthe use of a code made up of a series of distinct optical pulses,to be demodulated via a correlation detector [11. Salehi, Brack-ett, Chung, Wei, and Kumar developed the optical orthogonalcodes (OOCs) [2-51, which yield low interference crosscorrela-tions and therefore perform well under the correlation detector.Brady and Verdd analyzed the performance of this system in aPoisson channel using computationallyefficient arbitrarily tightbounds in [6]. Contributions have also been made by Lam andHussain who analyzed the correlation detector using momentspace bounds and Gaussian approximations to obtain estimatesof the probability of error for OCDMA systems using this de-tector [7]. Promising recent work by Mandayam and Aazhangprovides a method of simulatingan OCDMA ystem using im-portance sampling echniques[SI. Verdfi consideredthe optimaldemodulation in [9]in which a method of implementing the de-tector in a suboptimal yet more efficient manner was presentedand appropriate performance bounds were derived.T h i s paper considers a direct sequence OCDMA system, inwhich the user symbols are encoded into a series of high poweroptical pulses, which are then incoherently detected at the re-ceiver. The degradations considered in this model are restrictedto the Poisson effects of the photodetectionprocess, additive op-tical intensity noise, and multiple access interference. The focus
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IEEE TRANSACT1802Receiver 1\ -I
Passive FiberMulhplexer '2, Receiver K-xLn/
Fig. 1 . Optical code division multlple access system. Each user encodesthe laser pulses to transmit one of two predetermined sequences. The siGnalprocessing at the receiver depends onthe type of detechonused. The symbolb k ,represents the detector's decision on the kth user's symbol for frame position 1of this paper is to present a new detector for such OCDMA sys-tems. The studies cited above have shown that although thecorrelation detector allows a high throughput due to the pos-sibility of all-optical processing, it performs poorly for mostsystems composed of many users, even for ideal codes. It hasalso been determined that the optimal demodulation schemeimposes a computational complexity on the detector that is ex-ponential in the number of users. The demodulation schemeproposed here is based on an idea first introduced for the radiofrequency CDMA system by Varanasi and Aazhang [lo, 111.The concept is to obtain an estimate of the multiple access inter-ference from the knowledge of the possible signature sequencesand to employ this estimate to more accurately determine whatsymbols the different users have sent. A multistage detectorcan then be obtained by repeating this process using the newestimate of the user symbols to form a new estimate of the inter-ference. The computational complexity that this demodulationscheme imposes on the detector is only linear in the numberof users, and it is therefore much more practical than the opti-mal detector. Nevertheless, the performance of this multistagedetector shows significant improvement over the correlation de-tector and, in some cases, approaches the performance of theoptimal detector.The paper unfolds as follows, SectionI1describes he generaldirect detection OCDMA system. Subsections further developthe three detection options considered: the correlation detector,the optimal detector, and the multistage detector. The systemperformance analysis is presented in Section 111. Finally, nu-mesicd results are included in Section IV, where a random codeassumption is adopted for simplification, and a conclusion isthen given in Section V.
11. SYSTEM DESCRIFTIONAn optical code division multiple access system allows manyusers to access the optical channel simultaneouslyby using cod-ing to differentiate between them. This paper considers a systemconsisting of K users, labeled I% = 1, . . .,K , each containingan information source and destination, as shown in Fig. 1. Thetransmission frame is of length 2L + 1and the information sym-bols are denoted by b k , ~ , here I = - L , . . . L is the locationof the bit in the frame. The symbols are assumed binary, with
b k , ~ { 0, 1}, and they are also assumed statistically indepen-dent from all other symbols. The bit rate T is common to allusers, with each user experiencing a delay of r k E [0, T ) ela-tive to user one. Without loss of generality, user one is chosenasthe desired user throughout this paper. Furthermore, the detec-
' IONS ON COMMUNICATIONS,VOL. 42, NO. 21314, FEBRUARY/MARCH/APRIL 199
tor is assumed in perfect synchronization with this user's signa(i.e., 7 1 = 0). The symbol output by the information source oeach user is associated with one of two predetermined binarsequencesCAP) nd a t ) , epresenting bk , l = 1 or b k , l = 0, respectively. The sequences have entries in { 0, A}, with X a constant. Each sequence is of fixed length J and of weight M f 'where the weight is defined as the number of non-zero sequencentries. The bit interval is thus divided into J chips of timdurationT, = T / , each representing a t ? , for j = 1, . . .,J .This paper considers an OCDMA system employing a diredetection optical channel, as illustrated in Fig. 1. Each user contains a laser transmitter and a photodetector (PD), and all usershare a common optical fiber channel. The laser is employein pulsed mode, with an optical intensity pulse shape #( t ) ssumed non-negative and of extent much shorter than T,. Thlaser is modulated to transmit the sequencea?') by means of tapped delay line [12]. For each bit of each user, M f ' ) pulseare sent corresponding to each a & ) = A, for j = 1 , . ., JThe photodetector converts the incident optical field to an electrical current proportional to the intensity of the field. Eacphoton incident on the photodiode triggers a photoelectron wita given probability called the quantum efficiency. Since thmain concern of this paper is the degradation due to multipaccess interference, the degradations due to additive Gaussianoise, component non-linearities, and finite bandwidth are noincluded in the system model. In this model, the major degradation caused by the optical channel is the Poisson character of threceived process due to photon detection. The output current othe photodetector can thus be modeled as a conditional Poissoprocess Nt,given the incident optical intensity r ( t ) ,which iturn acts as the intensity process of the Poisson process.
The sample-function density for a Poisson process dependon the arrival times of photons as well as the total photon counIn practice, the exact arrival times cannot be acquired sincthe photon arrivals occur faster than any electronic equipmencan measure. The only statistics available at the receiver ara series of photon counts on predetermined intervals. Thistudy assumes that the count statistics are gathered over the chiintervals T,. Since the value of T, is assumed much largethan the extent of the optical pulse d ( t ) %he chip trmslsitioncan be assumed common to all users, with each optical pulsalways contained entirely within one chip interval. The systemthus appears chip synchronous at the detector, even though eacuser is not required to synchronize chip transitions with othemodeling the received signal can be equivalently modeled ahaving a rectangular pulse shape of duration T, instead of thactual pulse shapeq5(t),within each chip containing a pulse. Thconstant amplitudeof the rectangular pulse is determined by thpower received from each user. Since all users are assumeto have equal received pulse power, this constant designated aA = & $( t ) t is proportional to the average arrival rate ophotons in one chip. Thus for each user, the signature sequencepreviously described as a series of pulses, can be representeagain as the binary sequencesdi! with chip values either 0 or APhoto-optic detectors experience a residual current in th
users. With these assumptions, the Poisson intensity proces
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1803RANDT-PEARCE AND AAZHANG OPTICAL CODE DIVISION MULTIPLE ACCESS SYSTEMS
absence of incident light, which is called the dark current, andis equivalent to a low intensity background radiation. This isadded to other background radiation and is then representedas a constant additive intensity labeled Ad . The total Poissonintensity process for the data packet of the OCDMA system canbe written asL J
KUg$TC(t - T - j - ) T C - Tk)]+ Ad , (1)
k = 2
where uf)? is the jth chip of user le's signature sequence cor-responding to data symbol b k , l E { 0, 1) in frame position I ,and I&=(t)s a unit rectangular pulse of width T,. Since chipsynchronicity is imposed at the detector, the delays Tk are allmultiples of T,. The expression for the Poisson process in-tensity function can be simplified by considering only the timeinterval [0 ,T ) or the demodulation of symbol b1,o for the de-sired user l in frame position I = 0. Denote the symbol b1,o asb l and the corresponding user-one sequences as CL;'). Definingthe vector = ( b k , l - l , bk,l), let gk-' be the length J vectorwith j th component being the unchanged kth user contributionto the chip interval [iT + ( j - 1)Tc,T + jTc) , .e., aiE') sa concatenation of the latter part of ci;'-l) and the first part ofgr ' ) '. This can be easily done since the chip transitions aretaken to be the same for all users. The vector ai") can thusbe thought as a new code sequence for user le over the symboltime period [IT, I $- 1)T). Then for frame position d = 0, themodel for the intensity of the received light and proportionallythe intensity of the Poisson process in (1)can be written as
(B
J7 q t )= c(uy]nTc(t j - 1)T,)+j = 1
k = 2i E { O , 1 } , t E [ O , T ) (2)
for the desired user symbol bl = i. The vector of photoncounts in each chip for frame position d = 0 is labeled ={N1, . . . N J } ,where Nj denotes the Poisson distributed countover [ ( j - l)Tc, T,) with instantaneous intensity d i ) ( t ) asin (2).A. Correlation Detector
The performance of the OCDMA system described abovedepends on the signal processing or detection that is performedat the signal destination. The conventional detector employedby OCDMA systems is the correlation detector [l-81. Thisdetector decides on the desired user's symbol based on statisticscorresponding o the correlation of the received signal intensity'Frames are assumed to b e transmitted in direct succession so that the tran-
) and&EL ) are well defined.ition sequences
and a copy of the two expected sequences, 8")nd forthe desired user. This detector can be physically implementedusing two tapped-delay lines as correlators, photodetecting hesecorrelation values, and then comparing he results. This detectorfor user one is equivalent to(3)j = 1
where 7 is the threshold determined from the a priori probabil-ities of the user symbols and the sequence weights M i o ) andMi ') . The detector chooses 61 = 1 if the sum in (3) is greaterthan 7, chooses b l = 0 if it is less that 7, nd makes a randomchoice of b l = 1 with probabilityp if the threshold is achievedexactly, where p is based on the a priori probabilities and thepulse counts.The correlationdetector became popular in OCDMA systemsdue to its theoretical optimality n single user systems and due tothe potential for all-optical processing. The optical processingallows the electronic speed to limit only the user bit rate, withoutlimiting the spread spectrum bandwidth. The disadvantageof acorrelation detector is that in multiple access systems where theuser codes are known, it ignores this information so it does nottake full advantage of the known structure of the system, yieldinga poor performance for systems catering to large numbers ofusers. This suggests a need for examining the characteristics ofthe optimal detector, which is the topic of the next section.B. Optimal Detector
The minimum probability of error detector obeys the max-imum likelihood ratio test. Considering the incident intensityas a random variable, depending on b k I for 1 = - , . . . ,L andk = 2, . . . K , he received signal can be described as a doublystochastic Poisson process for each possible desired user sym-bol, bl = 0 and 61 = 1. The log likelihood function can bedescribed in two ways: by the method of conditioning and bythe method of self exciting point processes [13]. Conditioningthe point process on the intensity leads to the optimal detectoralgorithm described by Verdd in [9]. Describing the signal asa self-exciting point process lead to an optimum detector of theform
l n A = - (dl)( t)- do)(t>>t -t
where d i ) ( t )s defined asd i ) ( t ) E[T(~)(~)IJV~,L T 5 v < t] i E { 0, l} ,
and Nu is the Poisson point process with intensity functionT@ ~ ) ( v ) ,hich here is the actual intensity and not the equivalentconstant chip value expression in (2). The value of the thresholdq is the logarithm of the ratio of the a priori probabilities of userone sending bl = 0 and bl = 1. For the symbol asynchronouscase the likelihood ratio test depends on the photon arrivals for
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO . 21314,FEBRUARYMARCWAPRL 1991804
al l symbols in the frame, since the symbols overlap and thusthey all affect d')(t).The optimal detector has been shown to have complexityat best exponential in I
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BRANDT-PEARCE AN D AAZHANG: OPTICAL CODE DIVISION MULTIPLE ACCESS SYSTEMS 1805
Optimalusing !(I)-ewtor forUser 19
be implemented by employing the result of detector (10)for allusers to form a new estimate of the interference f k , j ( s ) usinga formula like (6) for iteration stage s + 1. The detector canthus attempt to reject the multiple access interference and yielda lower probability of error for all users. The iteration processis illustrated in Fig. 2.The computational complexity of the multistage detector foreach user is linear in the number of users. Each stage requiresKdetectors detecting the current frame positionuser symbols. Theprevious frame position symbols are also needed, yet they havealready been generated for the previous frame position calcula-tion. Thus, for S stages, there are a total of S K comparisonsrequired to estimate the symbol sent by one user. If the receiversare collocated, all users can use the same previous stage detectorestimates and the total K-user system complexity is on the orderof SIC. If the users are not collocated, the total complexity ofthe system is SK2.
111.PERFORMANCE NALYSISThis section presents methods of determining estimates ofthe average probability of bit-error for the OCDMA systemspresented above, for which numerical results are included inSection IV. Throughout this section, for computational simplic-ity a symbol synchronous system is assumed, i.e., all user delays
r k are assumed zero '.The correlation detector (3) and the multistage detector (9)both depend on a test statistic 2 of the form 2 = xi=l jfor some random vectorX. he vector X is a function of theactual interference level I = E:==,&+ A d (which may inThis statistic 2 is compared to a threshold to yield an averageprobability of error of the formitself be random), since Nj is Poisson with intensityala)j + Ij .
F e ( ~ ) q P r ( Z > qlbl = O,L) +T1 (2 < V l h = 1,L)+ epr (2 VlZ) (1 1)
where ~i = Pr(bl = i ) and e = p r o + 1 - p)nl is the proba-bility of error due to randomization if the threshold is achievedexactly given the randomization probability is p. The argument3Simulationresults of the asynchrono uscase ha ve shown virtually identicalperformance trends as the synch ronouscase.
of p e ( I ) ignifies that the error probability is given for one par-ticular realization of the actual interference levelr. The overallaverage probability of error requires an average over all users,i.e., over all possible combinations of interference levels. Thecalculation of the probability of error in general requires a J -fold convolution if the calculation is done directly. To diminishthe computationalcomplexity, the characteristic function can beused, as was first suggested for radio frequency CDMA sys-tems by Geraniotis and Pursley in [14] and later employed byAazhang and Poor in [151. The application of this technique toexpression (11) yields
_P,(L)= To F-l(@zlal=o,&))+O l l
TI F-l(@z~b,=l,&J))e +(%I&)), (12)z
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1806 IEEE TRANSACTIONS ON COMM UNICATIONS, VOL . 42 , NO. 21314, FEBRlJARYMARCHlAPRlL 1
X j takes at most (2K+ 1)different distributions, depending onthe values for Ij .B. Optimal Detector Analysis
A complication exists in the exact calculation of the proba-bility of error associated with the optimal detector P J o P t ) ,incethe expression depends on the expected value of the receivedintensity for all chips. The calculation of this expected value isof exponential complexity. Nonetheless, the probability of errormay easily be bounded, which is the approach developed in thissection.An upper bound to the error probability of the optimal detec-tor is trivial since any other detector that relies on no additionalinformation, such as the correlation detector and the multistagedetector, suffers an error probability higher than that of the opti-mal detector. A lower bound to the optimal detector error prob-ability can be found by considering what is referred to here asthe known-interferencedetector. The known-interferencedetector is defined by (4) except that the estimated intensitylevels d i ) ( t ) re replaced by the actual intensity levels d i ) ( t )for both i 1 and i = 1. This detector can obviously not beimplemented since the actual user symbols are not known at thedetector, yet its probability of error lower bounds the probabilityof error of the optimal detector. This is shown in the followingway.The likelihood function given that the exact interferenceisknown is equivalent to the likelihood function of a single userinterval [1T + ( j - l)Tc,ZT + jTc) .The likelihood ratio fo rdetecting symbol b1,o yieldsoptical system with dark current equal to I j 0 + A d in each chip
l n A = - ( d ) ( t ) d o ) ( t ) ) l tISince r(O)(t)= d ) ( t )outside the interval [0, Y), the detec-tion algorithm considers only the counts corresponding to thesymbol in question. The detector defined by the likelihood ra-tio determines the minimum probability of error detector, andtherefore yields a lower error probbility than a detector using
This lower bound for the optimal detector error probabilityis useful since it is easily calculated. The known-interferencedetector simplifies to an expression identical to (9) with Ij in-stead of fj . Therefore, (12) may be used with
i q t ) .
The characteristic function in this case is given by
j = 1
which is used in (12) to compute the probability of error. Nothat the characteristic unctioncan be employed in this case sinstatistical ndependence exists between the Poisson random vaables Nj and their coefficients (a!:; - ai:!) In (y)n Xgiven the interference evelL. The average error probability cagain be obtained by computing the average 1, = E1[Pe(The results of this are included in Section IV.C. Multistage Detector Analysis
It is easier to examine the statistics of the multistage detectof Section 11-C if one defines an arbitrary multistage detector
where the J-dimensional random vectorA(s) E [-I+ A d ,K1- I+ Ad ] is the only parameter changing between iterationFor the multistage detector defined by (9), the random vectA(s) used is the difference between the estimated interferenand the actual interference. written as
-( s ) = &s - 1) - I . (1The probability of error of the arbitrary multistage detectis a function of several random variables, namely N ,a(.), an
-. Throughout this section, he notationP,(& A(s),I) ignifithe probability of an errorof the s-stageof detection as a functiof the arguments, and averages of the probability of error aindicated by an over-bar and the omission from theP, rgumelist of the random variable used in averaging.The probability of error of the proposed multistage detectis nearly impossible to compute exactly due to the dependenof A(s) in (17) onE. Given a particular combination of codused, the exact probability of error for the ss stage of the actumultistage detector can be written as
where P, (14,&( ) L ) is the probability of an error ifA(s ) = S(is used as the interference estimate error in detector (16), if thinterference s actually Land the observed counts aren. Giventhe random vectorA( s ) as defined in (17) is uniquely determineby the counts8 inceN determines the symbol decisions othe previous stage giving rise to f(s - 1). The probability error can therefore also be written asP e ( 1 ) =r: Pe(n,S(s), I)P~(s)lN-,l(S(s>In,)Pjvl,l(I&lL
(1&(-(SIwhere the vector i ( s ) s in [-L + A d , K - 1- + Ad] and tprobabilityP~(s)ljv,I(&(s)124,) is unity for&(s)= l ( s- 1)-and zero otherwise.An approximation for the probability of error of the mulstage detector can be obtained by artificially mposing statisticindependence between the vector A(s) and the vector E. Thcan be done by simply using the marginal pA(,)lr(S(s>lL)
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BRANDT-PEARCE AN D AAZHANG: OPTICAL CODE DIVISION MULTIPLEACCESS SYSTEMS 1807
CZ LP h ( ~ ) i g , ~ ( ~ ( s )% L ) ~ j y r ( n / I )nstead of the conditionalprobabilityP~(s)lN,r(S(s)In,) in (191,pe (L> pe(fi,(s), ~ ) ~ ~ l ~ ( ~ I ~ ) ~ h ( s ) l ~ ( ~ ( s ) I ~ )
&(SI 14P&). (20)
Although it cannot be said that such an assumption of indepen-dence is in any way valid, the effect averaged over all possiblecounts is very close to that of the actual error encountered, asis shown when this approximation is compared to simulationresults in SectionIV. ntuitively this approximation s expectedto overestimate the error probability in general since the ac-tual detector uses the information in the counts8 o determineA(s) instead of making a random choice as calculated in theapproximation.The resulting approximation o the probability of error can becomputed using the characteristic function method in (14) sincethe chip counts and the weights are now independent given theinterference levelsr. The characteristic function in this case is
The calculation of p ~ ~ s ~ l ~ ( & ( s ) l ~ )n general requires a convo-lution of probability densities of the errors encountered by eachinterfering user. These densities can be immediately derivedfrom the error probabilities of the previous stage of the multi-stage detector.Iv. NUMERICALESULTS
This section presents the numerical results of the perfor-mance analysisof the OCDMA system described in the preced-ing section. The probability of error of the multistage detector isshown by comparing its approximation n (20)to the probabilityof error of the optimal detector and the correlation detector. Theerror probabilityof the optimal detector is bounded below by theerror probability of the "known-interference" detector as givenin Section 111-B and bounded above by the error probability ofthe multistage detector.Some additional assumptions have been made to simplifythe calculation. All user sequences are assumed random binarycodes, withPr(a [ ; = A) =n/andPr(atj. = 0) = 1 -A?/ Jwhere h;r = E I [ M , ( " )rJ , so that a sequence independent com-parison can be-made between the three detectors. This alsofacilitates the computationof expressions (15) and (21)by mak-ing the chip interference values independent and identically dis-tributed, thus allowing the total probability density to be writtenas a power of marginals. The marginal density of the interfer-ence errorp ~ ( ~ )an be easily calculated with this assumption. Inaddition, it assumed that all users have an equal a priori prob-abilityof sending each symbol, and that the detection thresholdis fixed at zero, independent of the random codes used.The baseline system used in all figures has the followingcharacteristics: the code length J = 300, the chip duration
lo-'a$kc , lo-*
E 1 0 . ~.3
97 -f- 2-~tage,K=S,Approximation:* -stage.K=5,Simulation. ---A- - 2-stage,K= 15,SimulationZ-stage,K= 15,Approximation- -& - -
1 0 . ~0 100J, code length 200
Fig. 3. Monte Carlo simulation results of the two-stage detector performancecompared to the analytic approximationfor a total numberof usersh = 5 and15. ThesystempmmetersareTc = 1 , X = 5 , Ad = O.l,and M / J = 0.05.
T, = 1,the laser pulse intensityX = 5 , he dark current intensityAd = 0.1 (A and )cd are both normalized to the number ofphotoelectrons generated per unit time), the average number ofpulses per symbol = .05 x J , and two examples of thetotal number of users K = 5 and 15. Each plot illustrates theperformance of the multistage detector system varying some ofthese parameters as marked.Monte Carlo simulation [8] results of the two-stage detectorsystem are presented to confirm the accuracy of the analytic ap-proximation. Fig. 3 compares the simulated probability of errorof the multistage detector system with the multistage detectorerror probability approximation for 5 and 15 users cases, as thecode length varies. The approximation s found to only slightlyoverestimate the simulated error probability in both cases, al-lowing for fluctuations inherent to simulation.The OCDMA system using a two-stage detector can supportmany users if the codes are sufficiently long, as illustrated byFig.'s 4 and5. The error probabilityof the system using the two-stage detector is uniformly lower than that of the system usingthe conventional correlation detector, and almost reaches thelower bound to the optimal detector for low correlation detectorerror probabilities, as is expected since the correlation detector isused to generate the first stage estimate of the interference evels.Fig. 4 illustrates the performance of the two-stage detector asthe code length J is varied, while in Fig. 5 , K is varied for twofixed values of J . The three detectors are equivalent for I( = 1,and the error probability decreases exponentially with the codelength in this case, as is expected for a Poisson system since thetotal received energy increases also.Since both the complexity and the performanceof the multi-stage detector increase with the number of stages, it is necessaryto determine how many stages are truly necessary to obtain asignificant improvement over the correlation detector. TWO rthree stages is sufficient in most cases, as evident in Fig. 6. Theimprovement beyond two stages is minimal for a small numberof users, yet becomes slightly higher for K = 15, when theinterference is the significant contributor to errors and the in-
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BRANDT-PEARCE A N D A A Z H A N G : OlIICAL CODE DIVISION MULTIPLE ACCESS SYSTEMS
0(c :O 10.~:3 :B$c i
22
1809
...............\\ K= 1 Irelation net .Ner Bound
*% 1...............-.. .......................................................I I0 5 10 15 20Pulse IntensityJ.
Fig. 8. Performance of the correlation and multistage detectors and lowerboundto the optimal detector as the pu lse intensity X increases fora total numberof users K = 1 and 5. The symbol X is given in number of photoelectronsp_er unit time. The s yste m parameters are J = 300, Tc = 1 , X d = 0.1, andM / J = 0.05.
5 3 ............... K = l,x 10.5% in-6- =S.Cor. Det.2 --r .K=15 Det., ,v , , , , ,K=S,Lower BoundK=15.Lower Bound- - -8 - =5.2-staaeDet. !lo - - -* K=15,2-stage Det. !10.001 .01 - . 1M/ J 1
Fig. 9. Performance of the correlation and m ultistage detectors and low erbound to the optimal detectoras the ratio of pulses to chipsa/ increases fora total number of users A = 1,5,and 15 . The sy stem parameters are J = 300,Tc= 1,X = 5, and Xd = 0.1.Undeniably, some of the effects in the previous figures aredue solely to that fact that random codes are employed. It is clearthat the performance of an OCDMA system with random codes
is worse on the average than a system with properly designedcodes that do not send pulses in the same chip and that usean optimal threshold. Using random codes imposes a flooron the error probability due to the possible equality betweencodes for symbols 1and 0. Fig. 10 illustrates the advantageofusing the multistage detector for a system with well designedTwo points are illustrated by Fig. 10, which shows a MonteCarlo simulation of an OCDMA system using OOCs as thepulse intensityX increases. First, the error probability decreasesexponentially as the pulse intensity increases, thus proving theeffects of random codes on the error probability plot in Fig. 8.
codes, namely optical orthogonal codes (OOCs)defined in [ 2 ] .
1.o 1.5 2.0 2.5 3.0 3.5Pulse Intensity, hFig 10. Simulation of the performance of an OCDMA system using opticalorthogonal codes with m aximum crosscorrelationof 2. Error probabili ties aresimulated for the correlation detector, the multistage detector, and the low erbound to the optimal detector as-the pulse intensity X varies. The systemparameters are J E 63 , K = 3, M = 9, Tc = 1 , and X d = 1 . The codes aregiven in [5 ,page 8701.
Second, the multistage detector is shown to provide a bettersystem performance than the correlation detector even for welldesigned codes.V. CONCLUSION
The OCDMA framework provides an opportunity to utilizethe vast bandwidth available on an optical channel. The systemconsidered in this paper is composed of K users, each trans-mitting binary encoded information simultaneously through anoptical channel. The multiple access interference, Poisson char-acteristic of the optical detection,and constant dark current noiseare the primary performance degradations considered. Descrip-tionsof the conventionallyused correlation detector and optimaldetectors are given.The main contributions of this study is the development ofa multistage detector for the OCDMA system, which is shownto have computationalcomplexity linear in the number of users,as compared to the optimal detector whose complexity is expo-nential in IC. This detector uses estimates of all user symbols togenerate an estimate of the multiple access interference, whichcan then be used to reject the interference. The estimates arederived without using photon arrival times that are necessary foroptimal detection but are usually not available. This process isiterated such that each stage of the detector employs the symboldecisions of the previous stage.Performance approximations and simulation results aregiven for this multistage detector. The multistage detector isshown to perform significantly better than the correlation de-Ad tested. The OCDMA system using the multistage detec-tor can support many more users than the conventional systemcould. The dependence of the error probability on the valueof total pulse energy and total number of pulses per symbol isgiven. Even as the number of stages S increases, the perfor-
tector in the random code case for all values of J,A,A, and
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mance keeps i nc reas ing , ye t the increase is marginal b e y o nd t w ostages. The re s u l t s p re s en ted also isola te the effects of th e ran-d o m c o d e a s s um p t i on . The pe r fo rmance of t h e s y s t e m d e v o i dof th i s a s s umpt ion is i l lus tra ted by a s i m u la t io n o f t h e O C D M As ys tem us ing low c ros s cor re la t ion codes .
In all cases , although t he m ul t i s tage de tec to r does no t per-f o r m s as wel l as t h e optimal detector , i t performs uniformlybe t t e r than the conventional correlation detector, particularlyfor l o w error probab i l i ty s ys tems , i.e., fo r s ys tems wi th a s ma l ln u m b e r o f u s e rs or l o n g c o d e l e n g t h s. The cos t of us ing themul t i s t age detector o v e r the corre la t ion detector is t he need fo re lec t ron ic s peed p roces s ing , which l imi t s the spread spectrumbandwid th . Ye t the ha rdware complex i ty fo r the mult is tage de-tec tor increases only l inearly in I(, ompared to the exponen t i a lcomplex i ty o f th e op t ima l de tec tor . S ince the complex i ty is onlylinear in the number of users a n d p h o t o n arrival time informa-
is not requ i red , the OCDMA system using the mult is tagede tec to r i s much easier t o im plement than the op t ima l de tec to r .
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Commun., vol. COM-37, no. 8, pp. 824-833, Aug. 1989.[3] J. A. Salehi and C. A. Brackett, Code division m ultiple-accesstechniques in optical fiber networks - Part 11: Systems perfor-mance analysis, IEEE Trans. Commun., vol. COM-37, no. 8 ,pp. 834-842, A ug. 1989.[4] E R. K. Chung, J. Salehi, and V. K. Wei, Optical orthogonalcodes: Design, analysis, and applications, IEEE Trans. on In-formation Theory,vol. IT-35, no. 3, pp. 595-604, M ay 19 89.[5j H. Chung and P. V. Kumar, Optical orthogonal codes - newbounds and an optimal construction, IEEE Trans. on Inform.?%eory,vol. IT-36, no. 4, pp. 866-873, July 1990.161 D. Brady an d S. Verd6 , A semiclassical analysis of optical codedivision multiple access, IEEE Trans. Commun.,vol. COM-39,no . 1, pp. 85-93, January, 1991.
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[13] D. L. Snyder, Random Point Processes. New York: John Wileand Sons, 1975.[14] E. A. Geraniotis and M. B. Pursley, Error probability fo r directsequence spread-spectrum multiple-access communicationspart ii: Approxim ations, IEEE Trans. Commun., vol. COM-30no. 5, p. 985-995,M ay 1982.[15] B. Aazhang and H. V. Poor, Performance of DS/SSMA commu
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Mart6 Brandt-Pearce(S91-M793 ceived a B.S. in Electrical Engineering and a B.A. in Aeplied Mathema tics from Rice University i1985, followed by a M.E.h. in 1989. Sh e worked with Lockh eed ESCin support of NASA Johnson Space Center from 1985 to 1989. DrBrandt-Pearce joined the faculty at the University of V irginia as an Assistant Professor of Electrical Engineering after c ompleting h er Ph.Din Electrical Engineering also from Rice University in 1993. H er current research interests include multiuser communication systems anoptical communication theory.
Behnaam Aazhang (S81-M85-SM91) was born in Bandar AnzaliIran, on Dec emb er7,1 957 . H e received his B.S. (with highest honors)M A , and Ph.D. degrees in Electrical and Computer Engineering fromUniversity of Illinois at Urbana-Champaign in 1981, 1983, and 1986respectively.From 1981 o 1985, he was a Research Assistant in the CoordinatedScience Laboratory, University of Illinois. In August 1985,he joinedthe faculty of Rice University, Houston, Texas, where he is now aAssociate Professor of Electncal a nd Computer Engineering. He wasVisiting Scientist at IBM Federal Systems Company, Houston, Texasin the Summ er of 1989. In the academic year 1991 -92 and in the Summer of 1993, he was with the Institut fbr Kommunikationstechnik aVisiting Professor. His research interests are in the areas of comm unication theory, information theory, and their applications with em phasion radio and optical multiple access communications; discrete evensimulations and perturbation analysis; and data driven nonlinear modeling.Dr. Aazhang is a recipient of the Alcoa Foundation Award 1993, thNSF Engineering Initiation Award 1987-1989, and the IBM GraduatFellowship 1 984-1985 , and is a member of Tau Beta Pi and Eta KappNu. He is currently serving as the Editor for Spread Spectrum Networkof IEEE Transactions on Communicationsand as the Secretary of thInformation Theory Society. He has also served as the PublicationChairman of the 19 93 IEE E International Sym posium on InformationTheory, San Antonio, Texas.
Swiss Federal Insthute ofTechnology (ETH) W d c h , Bwitzerland as